Symplectic Geometry and Isomonodromic
Deformations
Philip Paul Boalch

Wadham College, Oxford

D.Phil. Thesis Submitted Trinity Term 1999

## Abstract

In this thesis we study the natural
symplectic geometry of moduli spaces of
meromorphic connections (with arbitrary order poles)
over Riemann surfaces.
The aim is to understand the symplectic geometry of the monodromy data
of such connections, involving Stokes matrices.
This is motivated by the appearance of Stokes matrices in
the theory of
Frobenius manifolds due to Dubrovin,
and in the derivation of the isomonodromic deformation equations
of Jimbo, Miwa and Ueno.

The main results of this thesis are:

An extension to the meromorphic case of the infinite dimensional
description, due to
Atiyah and Bott, of the symplectic structure on moduli
spaces of flat connections.
This involves using an appropriate notion of singular $C^\infty$
connections and realises the natural moduli space of monodromy data
as an infinite dimensional symplectic quotient.

An explicit finite dimensional *symplectic*
description of moduli spaces of
meromorphic connections on trivial holomorphic
vector bundles over the Riemann sphere.
A similar description is given of certain extended moduli spaces
involving a compatible framing at each pole; these are the
phase spaces of the isomonodromic deformation equations.

A proof that the monodromy map is a symplectic map.
In other words the above two symplectic structures are related by the
transcendental map taking meromorphic connections to their
monodromy data.
The analogue of this result in inverse scattering theory is well-known
and was important in developing the quantum inverse scattering method.

A symplectic description of the full family of
Jimbo-Miwa-Ueno isomonodromic deformation equations.
In modern language we prove that the isomonodromic
deformation equations are equivalent to a flat *symplectic*
Ehresmann connection on a symplectic fibre bundle.
This fits together, into a uniform framework,
all the previous results for the
six Painlevé equations and
Schlesinger's equations.

Finally we look at the simplest case involving Stokes matrices in
detail.
We present a conjecture relating Stokes matrices to Poisson-Lie groups
(which we prove in the simplest case) and also
prove directly that in low-dimensional cases
the Poisson structure on the local moduli space of semisimple
Frobenius manifolds does arise from a Poisson-Lie group.

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